What is a Surd?

A root that cannot be written as a whole number or simple fraction. e.g. √2.

Multiplying Surds

Formula: √a × √b = √ab

Example: √2 × √3 = √6

Dividing Surds

Formula: √a ÷ √b = √(a / b)

Example: √10 ÷ √2 = √5

Squares and Roots

Squaring a square root cancels it out!

Example: (√7)² = 7

Simplifying Surds

Find a square factor (4, 9, 16, 25...) and take its root out.

Example: √75 = √(25 × 3) = √25 × √3 = 5√3

Adding/Subtracting

You can only combine "like" surds. 5√2 + 3√2 = 8√2.

Rationalising Denominators

Multiply top and bottom by the root to remove it from the bottom.

Example: 1 / √2 = (1 × √2) / (√2 × √2) = √2 / 2

Expanding Surd Brackets

Use FOIL just like algebra.

Example: (2 + √3)(2 - √3) = 4 - 2√3 + 2√3 - 3 = 1